超越Alice和Bob：改进的CONGEST中最大独立集的不可逼近性
Beyond Alice and Bob: Improved Inapproximability for Maximum Independent Set in CONGEST
摘要：
到目前为止，显示CONGEST模型下限的最有成效的技术是减少两方通信的复杂性。这种技术已经为各种基本问题，如距离计算，最小生成树，最小顶点覆盖等，产生了几乎紧密的结果。在这项工作中，我们采取这种技术更进一步，我们引入了一个框架，减少$t$ -党的通信复杂性，为每一个$t\geq 2$。我们的框架使我们能够显示最大独立集的改进的硬度结果。最近，Bachrach et al. [PODC 2019]使用两方框架来显示最大独立集近似的困难性。他们表明，找到一个$(5/6+ε)$ -近似需要$Ω(n/\log^6 n)$轮，找到一个$(7/8+ε)$ -近似需要$Ω(n^2/\log^7 n)$轮，在CONGEST模型中，$n$在网络中的节点数。我们改进了Bachrach等人的结果。通过减少多方通信的复杂性。我们的结果是：（1）在CONGEST模型中找到最大独立集的$(1/2+ε)$ -近似的算法需要$Ω(n/\log^3 n)$轮. (2)在CONGEST模型中找到最大独立集的$(3/4+ε)$ -近似的任何算法都需要$Ω(n^2/\log^3 n)$轮。
英文摘要：
By far the most fruitful technique for showing lower bounds for the CONGEST model is reductions to two-party communication complexity. This technique has yielded nearly tight results for various fundamental problems such as distance computations, minimum spanning tree, minimum vertex cover, and more. In this work, we take this technique a step further, and we introduce a framework of reductions to $t$-party communication complexity, for every $t\geq 2$. Our framework enables us to show improved hardness results for maximum independent set. Recently, Bachrach et al.[PODC 2019] used the two-party framework to show hardness of approximation for maximum independent set. They show that finding a $(5/6+ε)$-approximation requires $Ω(n/\log^6 n)$ rounds, and finding a $(7/8+ε)$-approximation requires $Ω(n^2/\log^7 n)$ rounds, in the CONGEST model where $n$ in the number of nodes in the network. We improve the results of Bachrach et al. by using reductions to multi-party communication complexity. Our results: (1) Any algorithm that finds a $(1/2+ε)$-approximation for maximum independent set in the CONGEST model requires $Ω(n/\log^3 n)$ rounds. (2) Any algorithm that finds a $(3/4+ε)$-approximation for maximum independent set in the CONGEST model requires $Ω(n^2/\log^3 n)$ rounds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
